# Math 570: Topology I

## Colorado State University, Fall 2017

**Instructor:** Henry Adams
**Email:** henry dot adams at colostate dot edu
**Office:** Weber 125
**Office Hours:** Tuesday 12-1pm and Wednesday 10-11am in Weber 125, or by appointment.

**Lectures:** MWF 9:00-9:50am in Weber 202
**Textbook:** *Introduction to Topological Manifolds* by John Lee (Second Edition).

An electronic copy of this book is freely available to CSU students here.

**Overview:** We plan to spend the first three weeks of the class on point-set topology (topologies, continuity, quotient spaces, connectedness, compactness). The goal for the rest of the course will be to introduce algebraic topology, while returning to topics from point-set topology as they arise. During the first third of the course, we hope to introduce the very basics of homotopy equivalences, the fundamental group, higher homotopy groups, and category theory. The second third of the course will focus on homology, with simplicial homology before singular homology. In the final third of the course we return to the fundamental group (Seifert-Van Kampen Theorem), covering maps, and other selected topics (some skipped along the way).

Though I hope to cover many of the topics in our textbook, I will cover some of them (such as portions of point-set topology, and the classification of compact surfaces) in a very casual way: I will provide intuition for the main theorems instead of giving complete proofs. Other topics, such as homology, will be emphasized and presented in detail.

**Syllabus:** Here is a course syllabus.

## Notes

Scans of Henry's lecture notes.## Homework

Homework 1 (LaTeX Source) is due Friday, August 25.Homework 2 (LaTeX source) is due Friday, September 1.

Homework 3 (LaTeX source) is due Friday, September 8.

Homework 4 (LaTeX source) is due Friday, September 15.

Homework 5 (LaTeX source) is due Friday, September 22.

Homework 6 (LaTeX source) is due Friday, October 6.

Homework 7 (LaTeX source) is due Friday, October 13.

Homework 8 (LaTeX source) is due Friday, October 20.

Homework 9 (LaTeX source) is due Friday, October 27.

Homework 10 (LaTeX source) is due Friday, November 10.

Homework 11 (LaTeX source) is due Friday, November 17.

Homework 12 (LaTeX source) is due Friday, December 1.

We will have weekly homework assignments. All homework is due in class at the beginning of class. Your homework should be legible and stapled.

## Exams

The exams will be in-class. You will only be able to use your brain and a pen or pencil - no notes, books, or electronic devices. Here is Practice Midterm 1.

Here is Practice Midterm 2.

Here is a Practice Final.

## Preparation Materials

One way to prepare for this course is by using the materials available at my class webpage Math 472, Fall 2016. Class notes, homework problems, and practice exams are available there. If you email me, I can also send you homework solutions and practice exam solutions. Not all of this material is assumed for Math 570, but we will be covering much of it at a fairly quick pace.## Schedule

Date |
Topic |
Remark |

Aug 21 | Course overview & Chp 1 | |

Aug 23 | Chp 2: Topologies, Convergence and Continuity | |

Aug 25 | Chp 2: Hausdorff Spaces, Bases and Countability | Homework 1 due |

Aug 28 | Class cancelled :( | |

Aug 30 | Chp 2: Manifolds | |

Sep 1 | Chp 3: Subspaces and Product Spaces | Homework 2 due |

Sep 4 | No class | |

Sep 6 | Chp 3: Disjoint Union Spaces, Chp 7: Categorical Products | Last day to drop or change grading option |

Sep 8 | Chp 7: Categorical Coproducts, Chp 3: Quotient Spaces | Homework 3 due |

Sep 11 | Chp 4: Connectedness | |

Sep 13 | Chp 4: Compactness | |

Sep 15 | Chp 4: Compactness | Homework 4 due |

Sep 18 | Chp 7: Homotopy | |

Sep 20 | Chp 7: The Fundamental Group | |

Sep 22 | Chp 7: The Fundamental Group | Homework 5 due |

Sep 25 | Chp 7: The Fundamental Group | |

Sep 27 | Review | |

Sep 29 | Midterm #1 **starting at 8:30am** | |

Oct 2 | Chp 7: Homomorphisms Induced by Continuous Maps | |

Oct 4 | Chp 7: Homomorphisms Induced by Continuous Maps | |

Oct 6 | Chp 7: Homomorphisms Induced by Continuous Maps | Homework 6 due |

Oct 9 | Chp 7: Categories and Functors | |

Oct 11 | Chp 7: Homotopy Equivalence | |

Oct 13 | Chp 7: Higher Homotopy Groups | Homework 7 due |

Oct 16 | Chp 5: Cell Complexes and CW Complexes | End of course withdrawal period |

Oct 18 | Chp 5: Simplicial Complexes | |

Oct 20 | Class cancelled; read Theorem 7.40 & Lemma 7.45 | Homework 8 due |

Oct 23 | Chp 13: Simplicial Homology | |

Oct 25 | Chp 13: Simplicial Homology | |

Oct 27 | Chp 13: Simplicial Homology | Homework 9 due |

Oct 30 | Chp 13: Singular Homology | |

Nov 1 | Review | |

Nov 3 | Midterm #2 **starting at 8:00am** | |

Nov 6 | Chp 13: Singular Homology | |

Nov 8 | Midterm 2 and Homework 10 comments | |

Nov 10 | Chp 13: Singular Homology | Homework 10 due |

Nov 13 | Chp 13: Functoriality of simplicial homology | |

Nov 15 | Chp 13: Difference between homotopy groups and homology | |

Nov 17 | Chp 13: The Mayer-Vietoris Theorem | Homework 11 due |

Fall Recess, Nov 20-24 | ||

Nov 27 | Chp 13: Homotopy invariance | |

Nov 29 | Chp 13: Homotopy invariance | |

Dec 1 | Chp 13: Mayer-Vietoris example | Homework 12 due |

Dec 4 | Chp 13: Degree theory for spheres | |

Dec 6 | Chp 13: Degree theory for spheres | |

Dec 8 | Review | |

Final Exam, Monday December 11 | ||

7:30-9:30am in Weber 202 |